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This analysis reveals full automorphism groups in symmetric designs, indicating their complex geometry.
It is clear that the full automorphism group of the $(15,8,4)$-design of points and hyperplane complements of PG(3,2) is GL(4,2). Using methods of point-line geometries, we determine the full automorphism groups of the remaining four symmetric $(15,8,4)$-designs and describe their actions on the sets of points and blocks.
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Pankov et al. (2025) studied this question.
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